{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $(X_n)_{n \\ge 0}$ be a sequence of real random variables, [[adapted|YRNFE016FF]] to a [[filtration|HNK1Y62G7V]] $(\\mathcal{F}_n)_{n \\ge 0}$, with $E[\\abs{X_n}] \u003c \\infty$ for all $n$.\nThen there are a [[martingale|0CDVK9RQBH]] $(M_n)_{n \\ge 0}$ and a sequence $(A_n)_{n \\ge 0}$ of integrable random variables with $A_0 = 0$ and $A_n$ $\\mathcal{F}_{n-1}$-measurable for $n \\ge 1$, such that\n$$X_n = M_n + A_n \\quad \\text{for all } n \\ge 0.$$\nThis decomposition is unique up to equality almost surely.\nExplicitly,\n$$A_n = \\sum_{k=1}^{n} \\bigl(E[X_k \\mid \\mathcal{F}_{k-1}] - X_{k-1}\\bigr).$$\nMoreover, $(A_n)$ is almost surely non-decreasing if and only if $(X_n)$ is a [[submartingale|EQ055QMACW]].\n\n## Remarks\n\nExistence: with $A_n$ as above, $M_n = X_n - A_n$ satisfies $E[M_n - M_{n-1} \\mid \\mathcal{F}_{n-1}] = E[X_n \\mid \\mathcal{F}_{n-1}] - X_{n-1} - (A_n - A_{n-1}) = 0$.\nUniqueness: if $M_n + A_n = M'_n + A'_n$, then $M - M'$ is a martingale that is predictable, so $M_n - M'_n = E[M_n - M'_n \\mid \\mathcal{F}_{n-1}] = M_{n-1} - M'_{n-1}$, and it is constant equal to $0$.\nExample: for the simple symmetric random walk $S_n$, $S_n^2 = (S_n^2 - n) + n$, so $A_n = n$.\nA sequence $(A_n)$ with $A_n$ $\\mathcal{F}_{n-1}$-measurable is called predictable.\nWithout the predictability of $A$ the decomposition is not unique: $X_n = 0 + X_n$ is another split into a martingale and an adapted process.\n\n## Explanations\n\n### L1 human\n\nAny random process with finite means splits into a fair-game part and a trend part.\nThe trend is the part you can predict one step ahead, and the fair game is the surprise.\nFor a submartingale the trend only goes up.\n\n### L2 human\n\nThe predictable part $A$ of $M^2$ for a square-integrable martingale $M$ is the predictable quadratic variation $\\langle M \\rangle$, a basic tool for martingale central limit theorems and stochastic integration.\nThe decomposition is the discrete-time model for the [[Doob–Meyer decomposition theorem|SX7ZPP72EM]], whose continuous-time proof is much harder.\nIt is also the discrete form of the split of a [[semimartingale|MZNFPJJMY9]] into a local martingale and a finite variation part.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "0D0GBNDKDH",
      "6M852B4RA8"
    ],
    "kind": "statement",
    "msc": [
      "60G42"
    ],
    "name": "Doob decomposition theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q1242398",
      "wikipedia": "Doob decomposition theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Doob_decomposition_theorem"
      }
    ],
    "tag": "YHSSQ67GDS",
    "uses": [
      "0CDVK9RQBH",
      "EQ055QMACW",
      "YRNFE016FF",
      "HNK1Y62G7V",
      "0YAMJRK04E"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/YHSSQ67GDS/",
  "links": [
    {
      "field": "about",
      "target": "0D0GBNDKDH"
    },
    {
      "field": "about",
      "target": "6M852B4RA8"
    },
    {
      "field": "uses",
      "target": "0CDVK9RQBH"
    },
    {
      "field": "uses",
      "target": "EQ055QMACW"
    },
    {
      "field": "uses",
      "target": "YRNFE016FF"
    },
    {
      "field": "uses",
      "target": "HNK1Y62G7V"
    },
    {
      "field": "uses",
      "target": "0YAMJRK04E"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YHSSQ67GDS.md",
  "msc": [
    "60G42"
  ],
  "name": "Doob decomposition theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YHS/YHSSQ67GDS.md",
  "status": null,
  "tag": "YHSSQ67GDS",
  "uri": "m2://YHSSQ67GDS",
  "url": "https://math2.tamnd.com/wiki/doob-decomposition-theorem/",
  "writing": {
    "tag": "YHSSQ67GDS",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}